Answer:
The bending stress of the face tooth is ![\sigma _{bg} = 502.82 MPa](https://tex.z-dn.net/?f=%5Csigma%20_%7Bbg%7D%20%3D%20502.82%20MPa)
Explanation:
From the question we are told that
The number of tooth of the pinion is ![N_t = 26 \ tooth](https://tex.z-dn.net/?f=N_t%20%3D%2026%20%5C%20tooth)
The velocity of rotation is given as ![\omega_p = 1800 rpm](https://tex.z-dn.net/?f=%5Comega_p%20%3D%201800%20rpm)
The number of tooth is of the gear is ![N_g = 55 \ tooth](https://tex.z-dn.net/?f=N_g%20%3D%2055%20%5C%20tooth)
The quality level is ![Q_r = 10](https://tex.z-dn.net/?f=Q_r%20%3D%2010)
The transmitted tangential load is
= ![22 KN * \frac{1000N}{1KN} = 22*10^3 N](https://tex.z-dn.net/?f=22%20KN%20%2A%20%5Cfrac%7B1000N%7D%7B1KN%7D%20%3D%2022%2A10%5E3%20N)
![k_m = 1.7](https://tex.z-dn.net/?f=k_m%20%3D%201.7)
The angle of the teeth is ![\theta_t = 20^o](https://tex.z-dn.net/?f=%5Ctheta_t%20%3D%2020%5Eo)
The module is ![M= 5](https://tex.z-dn.net/?f=M%3D%205)
The face width is ![W_f = 62mm](https://tex.z-dn.net/?f=W_f%20%3D%2062mm)
The diameter of the pinion is mathematically represented as
![d_p = M * N_t](https://tex.z-dn.net/?f=d_p%20%3D%20M%20%2A%20N_t)
Substituting the values
![d_p = 5 *26](https://tex.z-dn.net/?f=d_p%20%3D%205%20%2A26)
![= 130 mm = \frac{130}{1000} = 0.130m](https://tex.z-dn.net/?f=%3D%20130%20mm%20%3D%20%5Cfrac%7B130%7D%7B1000%7D%20%3D%200.130m)
The pitch line velocity is mathematically represented as
![V_t = \frac{d_p }{2} \frac{2 \pi \omega_p}{60}](https://tex.z-dn.net/?f=V_t%20%3D%20%5Cfrac%7Bd_p%20%7D%7B2%7D%20%5Cfrac%7B2%20%5Cpi%20%5Comega_p%7D%7B60%7D)
Substituting values
![= \frac{0.130}{2} * \frac{2 * 3.142 * 1800 }{60}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B0.130%7D%7B2%7D%20%2A%20%5Cfrac%7B2%20%2A%203.142%20%2A%201800%20%7D%7B60%7D)
![= 12.25\ m/s](https://tex.z-dn.net/?f=%3D%2012.25%5C%20%20m%2Fs)
Generally the dynamic factor is mathematically represented as
![K_v = [\frac{A}{A +\sqrt{200V_t} } ]^B](https://tex.z-dn.net/?f=K_v%20%3D%20%5B%5Cfrac%7BA%7D%7BA%20%2B%5Csqrt%7B200V_t%7D%20%7D%20%5D%5EB)
Now B is a constant that is mathematically represented as
substituting values
![= \frac{(12- 10 )^{2/3}}{4}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B%2812-%2010%20%29%5E%7B2%2F3%7D%7D%7B4%7D)
![=0.3968](https://tex.z-dn.net/?f=%3D0.3968)
A is also a constant that is mathematically represented as
![A = 50 + 56(1 -B)](https://tex.z-dn.net/?f=A%20%3D%2050%20%2B%2056%281%20-B%29)
Substituting values
![= 50 +56 (1- 0.3968)](https://tex.z-dn.net/?f=%3D%2050%20%2B56%20%281-%200.3968%29)
![= 83.779](https://tex.z-dn.net/?f=%3D%2083.779)
Substituting these value into the equation for dynamic factor we have
![K_v = [\frac{83.779}{83.779 + \sqrt{200 * 12.25} } ]^{0.3968}](https://tex.z-dn.net/?f=K_v%20%3D%20%5B%5Cfrac%7B83.779%7D%7B83.779%20%2B%20%5Csqrt%7B200%20%2A%2012.25%7D%20%7D%20%5D%5E%7B0.3968%7D)
![= 0.831](https://tex.z-dn.net/?f=%3D%200.831)
The geometric bending factor for a 20° profile from table
"AGMA Bending Geometry Factor J for 20°, Full -Depth Teeth with HPSTC Loading , Table 2-9"
That corresponds to 55 tooth gear meshing with 26 pinion is
![J_g = 0.41](https://tex.z-dn.net/?f=J_g%20%3D%200.41)
the diameter pitch can be mathematically represented as
![p_d = \frac{1}{M}](https://tex.z-dn.net/?f=p_d%20%3D%20%5Cfrac%7B1%7D%7BM%7D)
Substituting values
![p_d = \frac{1}{5}](https://tex.z-dn.net/?f=p_d%20%20%3D%20%5Cfrac%7B1%7D%7B5%7D)
![=0.2mm^{-1}](https://tex.z-dn.net/?f=%3D0.2mm%5E%7B-1%7D)
The mathematically representation for gear tooth bending stress in the teeth face is as follows
![\sigma_{bg} = \frac{F_T \cdot p_d }{W_f * J_g}\frac{K_a K_{dt} }{K_v} K_s K_B K_t ----(1)](https://tex.z-dn.net/?f=%5Csigma_%7Bbg%7D%20%3D%20%5Cfrac%7BF_T%20%5Ccdot%20p_d%20%7D%7BW_f%20%2A%20J_g%7D%5Cfrac%7BK_a%20K_%7Bdt%7D%20%7D%7BK_v%7D%20K_s%20K_B%20K_t%20----%281%29)
Where
is the tangential load
is the face width
is the application factor this is obtained from table "Application Factors, Table 12-17 " and the value is
= 1
is the load distributed factor
is the size factor
is the rim thickness factor which is obtained for M which has a value 1
is the idler
Substituting values into equation 1
![\sigma_{bg} = \frac{22*10^3 *0.2}{62 * 0.41} * \frac{1 * 1.7 }{0.831} * 1 *1 *1.42](https://tex.z-dn.net/?f=%5Csigma_%7Bbg%7D%20%3D%20%5Cfrac%7B22%2A10%5E3%20%2A0.2%7D%7B62%20%2A%200.41%7D%20%2A%20%5Cfrac%7B1%20%2A%201.7%20%7D%7B0.831%7D%20%20%2A%201%20%2A1%20%2A1.42)
![= 502.82 N/mm^2](https://tex.z-dn.net/?f=%3D%20502.82%20%20N%2Fmm%5E2)
![= 502.82 * 1000 * \frac{N}{m^2}](https://tex.z-dn.net/?f=%3D%20502.82%20%2A%201000%20%2A%20%5Cfrac%7BN%7D%7Bm%5E2%7D)
![= 502.82 MPa](https://tex.z-dn.net/?f=%3D%20502.82%20MPa)